Paper ID #26527

Toward a T-Shaped Integration of Mathematics in Mechanical Engineering

Dr. Amitabha Ghosh, Rochester Institute of Technology

Dr. Amitabha Ghosh is a licensed Professional Engineer with a Ph.D. in general engineering composite(Major: Aerospace Engineering) from Mississippi State University. He obtained his B.Tech. and M.Tech.degrees in Aeronautical Engineering from Indian Institute of Technology, Kanpur. He is a professor ofMechanical Engineering at Rochester Institute of Technology. His primary teaching responsibilities are inthe areas of fluid mechanics and aerodynamics. He is also a significant contributor in teaching of the solidmechanics courses. For the past ten years, he has been involved heavily in educational research at RIT andhas also served as the coordinator of the Engineering Sciences Core Curriculum (ESCC) in MechanicalEngineering.

c©American Society for Engineering Education, 2019

Toward a T-Shaped Integration of Mathematics in Mechanical Engineering

Abstract

This paper presents a progress report structured to implement instructional methods presented in

3 earlier papers published by the author. Details of the coordinated instructional and assessment

approaches were utilized by a faculty team in an engineering sciences core curriculum (ESCC)

and are now extended to some upper level technical electives. These instructional guidelines

have been part of the ABET continuous improvement process at the author’s institution.

After the release of ABET 2000, various teaching and assessment methods have been explored

by different faculty teams for engineering science topics. Their recommendations were

implemented to develop pedagogy and assessment in ESCC. However, these courses failed to

uniformly reinforce important mathematical concepts for various reasons. This shortfall was

cited and discussed in recent publications. It is necessary to determine which effective

mathematical tools are most needed to teach formulation and solution skills to mechanical

engineering students, and how to train engineering faculty to use such tools.

Recently the departmental focus also shifted to topics of applied mathematics necessary to lead

students into advanced research in continuum mechanics. Our initial attempt was to choose the

most important mechanical engineering topics and demonstrate conceptual breadth and depth

necessary for connectivity with previous topics. Last year a status update was published. This

paper further presents a flowchart of mathematical preliminaries and their connectivity to

advanced fluid mechanics. The examples presented here demonstrate student performance

improvement in topics of aerodynamics and ideal flows. The focus group of students

demonstrated remarkable clarity in expressing logical arguments. The author believes such

recommendations should further be explored and implemented for other solid mechanics and

continuum mechanics electives. The current results support a T-shaped integration of applied

mathematics in mechanical engineering. They can be shown to link mathematical training of

engineering students and desirable ABET outcomes.

Introduction

A few years ago, the college of engineering at Rochester Institute of Technology (RIT)

appointed a new dean. The new dean added a new vision: Build a T-Shaped Curriculum. Few

knew what it meant exactly. So, faculty in the college defined and accepted the vision as an

axiom. The shape of the letter T illustrates the structure of such a curriculum – providing breadth

(by the horizontal arm) and depth (by the vertical leg) of student learning in all engineering

disciplines. The ESCC team in mechanical engineering (ME) had already designed an effective

core engineering curriculum almost a decade before this time. It had to make changes according

to this new focus. The effort in the present paper is to discuss the role of mathematics for

implementation of such a T-shaped curriculum.

ME students learn a significant amount of applied mathematics to succeed functionally. How can

the presentation style of conventional mathematical topics be improved so that students become

better learners, and also retain mathematical thoughts for life? This is the research focus now.

We present an archived multiple choice (MC) examination question to begin discussion.

Fig.1 Student performance assessment example from a Dynamics final examination

The above question was published in an assessment study [1] a few years ago by the Dynamics

faculty of ESCC. The examination area involves integration of the acceleration function over a

space variable. Most students with wrong answers chose option (E) instead of the correct choice

(C). Lack of student performance on this and other examples quoted on the study created a new

focus area for faculty to rectify. The deficiency area on this question had already been identified

before (and instructional focus had already been implemented by ESCC), and yet there was a

further twist in assessing student performance due to a new type of deficiency [2] identified in

that particular study.

In 2016, another paper was published by the ESCC group which suggested remedial actions in an

introductory/remedial statics course [2]. A follow-on report was published last year [3] which

focused further on the remedial actions for specific mathematical deficiencies. The ESCC focus

earlier had been to determine connectivity of common dynamics topics which would allow better

retention of mathematical concepts. At that time, interest was not focused on details of

mathematical connectivity in the whole ME curriculum. This paper presents an expanded

implementation of recommendations presented in [3]. We have now broadened connectivity of

specific mathematical topics necessary in the whole ME curriculum beginning with high school.

The examples illustrated here are suitable for getting students ready for aerodynamics or,

computational fluid dynamics (CFD). We gathered test results for assessing our approach on a

small segment of ideal flow topics presented in an upper level undergraduate course in fluid

mechanics. Similar connectivity may be established for other technical electives.

Since 2006 ME faculty at RIT have systematically compiled examinations and performance data

for ESCC use. Teams of faculty who participated in assessing performance on these

examinations had developed a seamless teaching/learning environment for ME students which

may be adapted to improve instructional strategies if necessary. When fully implemented, these

help ME students learn better and meet ESCC direct assessment goals measured by their

examination performance. A process for continuous improvement of the curricular design has

been implemented in ESCC. The breadth of the RIT course offerings and experiential learning,

together with the depth of upper level elective classes provide the RIT mechanical engineers a T-

shaped integration of mathematical concepts. Since conceptual assessment is largely based on

student responses on MC type examination questions, a brief discussion of MC performance

analysis is necessary. Interested readers are requested to find more details in publications [4 – 7].

MC Assessment Process

ESCC faculty teams have studied causes of common failures on examinations using sets of

carefully designed MC questions for several years. ESCC collects and preserves actual final

examinations and student answers to both MC and analysis questions to develop a deeper

understanding of the causes of failure. All ESCC courses have common home assignments and

examinations. Some MC questions are always posed in addition to detailed analysis questions on

examinations. The hypothesis is: MC questions, when posed correctly, can accurately determine

areas of conceptual difficulties and analysis questions may further provide supporting evidence

for such claims. Monitoring grading uniformity on analysis questions in large sections taught by

several different instructors is very difficult. So, student performance on MC questions is always

considered first.

Figure 2 shows analysis of 4 questions from ESCC faculty archives to clarify the process. In

ESCC classification, a “well-posed” MC question has three requirements. The question must be

1) clearly written, 2) error-free, and 3) answerable within 3 minutes of testing time for average

students. Faculty are asked to focus on one or two key concepts only to design the question.

Otherwise the question is not posed as an MC question.

Category A questions are those in which questions are well-posed, and 60% or more of the class

can answer them correctly. On figure 2, Q1 and Q4 fit this category. Category B questions are

those where the questions are well-posed but less than 60% of the class can answer them

correctly. Here Q2 fits that category. The response in Q3 on the other hand shows a completely

different trend. Such responses may happen due to one of three reasons: 1) the question is

incorrectly posed (viz. choices are ambiguous, figures are unclear, expressions/equations have

errors, solutions are too long to obtain in the allotted time, etc.) or, 2) incorrect solutions are

marked for processing (indicating a human error), or, 3) many students do not answer that

question. The first two reasons are rectifiable. In figure 2 above, the reason 2) was applicable.

After receiving the scantron report, the human error was quickly discovered and corrected before

the examination grades were posted. Ill-posed questions are initially marked category C (needing

corrections). After scrutiny and corrections, such questions may finally be considered in category

A again.

Over the years ESCC faculty have learnt to design the answer choices on an MC question in such

a way that reveals weak performance due to a specific reason. For example, choices D and E in

Q2 have some such clues which would be explored further and corrective instructional strategies

would be developed. For this purpose, the work space given for each MC question is collected

after the test in addition to the answer sheet. Today, archived examination data are available for

comparing test performance (on the same/similar question) on midterm examination with final

examination, and performance on final examinations over the years. In the present paper we

analyzed performance of both MC and analysis type questions on ideal flow topics. Additional

description of instructional methods and delivery ideas may be found in references [2] and [3].

After this brief introduction of our assessment process, we now begin searching for mathematical

concepts connecting ideal flows.

Mathematics as an Integrator

Mathematical physics has been the catalyst of all engineering advancements for mankind.

Beginning with Newton all analytical thoughts found their basis in this subject until computer

age began in the 20st century. Easy availability of powerful computational solutions detracts from

mathematical thought development today. The purpose of this paper is to initiate a renewed

focus on logical reasoning with emphasis leading to connectivity of relevant mathematical

topics. We offer suggestions for optimized learning and retention of mathematical thoughts

which, we believe, must be practiced by mechanical and aerospace engineers.

Ideal Flows – Pedagogical Relevance

Traditionally ideal flows require classical methods of solution plus powerful complex analysis

ideas such as complex potential, complex velocity, complex force, and the residue theorem. But

the course on which ideal flows data was collected is an undergraduate elective course in fluid

mechanics (Fluids II), and our ME students are not required to take a pre-requisite course

introducing complex analysis. Since all students do not have the same background, only classical

solution methods are taught. This makes the current course harder to teach. But employing three-

course connectivity of topics before on a graduate level had proven to yield better student

learning [4]. The same idea was replicated now at the undergraduate level with support from

assessment data [3, 7]. As a result of these efforts, students in the present data pool benefited.

Student teams demonstrated better reviewing methods, and individual students demonstrated

better performance on class tests. The traditional lecture delivery was often flipped for enhanced

learning in groups. Brainstorming was encouraged with modified assessment methods as

explained later.

MC test questions (and occasionally analysis questions) were used to provide connectivity of

topics (see figures 5 – 7 given later). Each of the questions used in this paper was actually posed

on a recent examination. Archived data show where and which course earlier introduced the

concept and where in the curriculum reinforcement was achieved. The next section presents three

steps of conceptual connectivity required in the present approach, followed by some results and

some new discoveries. Finally select recommendations are presented and concluded.

Establishing conceptual connectivity – Step 1

Since ideal flows is an essential topic to be understood by all mechanical engineers who wish to

master the fluid mechanics area, this area was selected to implement the methodology.

Continuity is tested tracing back to the high school and early college courses in mathematics.

The reader may consider these recommendations and implement any remedial strategies

contextual to his/her own university. This work is considered a developing field where focus has

become necessary in recent times. Necessary data which have been the focus for corrective

actions before would be recalled here and further connected with upper level conceptual

understanding of mathematics. Although this is a study to enhance educational research, faculty

involved with fundamental research in classical aerodynamics (or ideal flows) would find this

work illuminating to understand mathematical bottlenecks experienced by average undergraduate

students in mechanical engineering today.

Week Text Sections Course Content: Topics

Covered

Home Work & Practice

Problems

HW, Test/Quiz

Coverage

08/27/18

through

08/31/18

Chapters 1 – 6

Reviews (Note:

Chapters 3 and

5 expand

topics of

MECE210

class)

3.1 – 3.3

Review of Fluid Mechanics and

Calculus, Control Volume (CV)

Approach, Flow Visualization

Tools, Streamlines, Pathlines &

Streaklines. Screening of the

Movie: Vorticity (parts I & II)

HW1: 1-47, 2-50, 3-16, 4-14, 6-

11

Practice : 1-43, 2-42, 3-10, 4-30,

6-11

Sign and return the

honor principle to

earn the HW0

credit.

Quiz 1 will test the

pre-requisite

concepts for this

class

09/03/18

through

09/07/18

3.4 – 3.5

Sept. 3 (Labor Day, No

Classes!)

Flow Acceleration

Read through Introduction.doc

on the 550CD to get

connectivity of all topics and

detailed derivations.

HW2: 3-5, 3-36, 3-38, 3-41, 3-43

Practice: 3-14, 3-18, 3-27, 3-35

HW1 due this week

Quiz No. 2 will test Summary Reviews

of Chapters 1 – 6

(see SummaryCh1-

6.pdf on

myCourses)

09/10/18

through

09/14/18

5.1 – 5.5

7.1 – 7.8

Euler's and Bernoulli equations

Differential Fluid Flows, Euler,

Bernoulli and the Navier-

Stokes equations, Stream

Function, Velocity Potential

HW3: 5-4, 5-9, 5-11, 5-21, 5-25

Practice: 5-19, 5-23, 5-26, 5-27

HW2 due this week

Quiz No. 3 will test

on the Movie topics

09/17/18

through

09/21/18

7.9 - 7.11

Examples of Solutions of

Laplace equation

Elementary Plane Flows

Superposition Principle

HW4: 7-4, 7-13, 7-15, 7-21, 7-24

Practice 7.6, 7-12, 7-17, 7-20,

7.26

HW3 due this week

Quiz No. 4 will test

Ch3 and 5 Concepts

09/24/18

through

09/28/18

8.1 – 8.5

Superposition Principle

(contd.) Introduction to

Aerodynamics and Wind

Tunnels; Dimensional Analysis

and Similitude Review

HW5: 7-61, 7-67, 7-70, 7-77, 8-

37

Practice: 7-47, 7-60, 7-73, 8-56,

8-60

HW4 due this week

Quiz No. 5 will test

Ch. 7 Concepts

Figure 3: Fluids II syllabus in part leading up to the ideal flow topics

Figure 3 shows a typical syllabus leading up to the topics of ideal flows presented in Fluids II or

any upper level second fluid mechanics class. The course begins with discussion of continuum

hypothesis, basic fluid properties and units. After the introductory discussion of fluid statics,

selected gadgets such as manometers, pressure gages and pitot tubes are reviewed. These topics

were first introduced in the first fluid mechanics course, but students must review them for

problem solving [8]. All homework and practice problems are from this reference textbook.

Kinematics discussion in fluid flows contrasts fluid behavior with rigid body dynamics and

introduces two additional topics of shear deformation and dilatation. The substantial derivative

representing fluid acceleration introduces both local and convective terms. These terms become

difficult to conceptualize for average students unless stream tubes and other CV examples are

used. Worked out examples, class notes, and complete derivations are provided on files archived

at the class site in myCourses. The movie Vorticity developed by A. H. Shapiro [9] is an

excellent learning aid for these topics. The movie is linked to open from the Internet on

myCourses and also screened and discussed in class. Both Lagrangian and Eulerian flow

descriptions are presented and contrasted. The class reviews the determination and sketching of

streamlines. Then stream function and velocity potential are introduced, and connected

conceptually to an orthogonal grid system seen later in CFD. The evaluation of partial

derivatives and integration of functions of several variables are presented in full rigor. After

reviewing dimensional analysis, Buckingham’s pi theorem and streamline coordinates, Bernoulli

equation is recalled. The added understanding due to Crocco’s theorem recalled from the movie

[9] relaxes the use of the application of Bernoulli equation between any two points in a flow field

rather than two points on a particular streamline.

Later topics in the course present a big conceptual leap for students. Do they possess sufficient

understanding to explain when solution of Laplace equation may replace the solution of Euler’s

equations? Have they understood ideal flow boundary conditions and pressure-velocity relations?

Can they integrate functions of several variables? If they can answer these questions well, they

should be mathematically prepared to learn ideal flow applications. We had earlier experienced

difficulty in the implementation phase because reinforcement of mathematics was difficult in a

single course to provide complete connectivity. Using the ideal flow concepts, we opened a

backward tracing approach taking us back to college and high school mathematics which will be

presented now in connectivity steps 2 and 3 below.

Mechanical engineers do not need to have the full mathematical rigor of aerodynamicists in ideal

flows. But students must receive adequate background to appreciate its scope in meteorology,

boundary layers and wind tunnel applications which emphasize the superposition principles. In

four weeks, Fluids II class offers an abridged introduction leading to CFD. The inverse design

problems and construction of Euler solvers in CFD require a complete understanding of

governing differential equations and boundary conditions. Linking the undergraduate

mathematical base requires reviewing some concepts first seen in high school. The next section

presents specific topics that assist understanding formulation and problem solving in this course.

Connectivity – Step 2

After identifying the technical topics presented in figure 3 it is clear that the review focuses on

both control volume analysis and differential equations. The relevant physical concepts link the

following mathematical topics with our approach (Fig. 4). The analytical methods require

mathematical concepts of Taylor series, line, surface and volume integrals, sign conventions of

surfaces and stresses, review of directional lumping and differential element choices for CV and

boundary conditions, Leibnitz rule, and integration by parts (depending upon problem

selections). Also reviews of basic algebraic identities (applicable for polynomials), functions of

several variables, partial differentiation, and various coordinate systems are necessary. Since

tensor representations are not possible (our ME undergraduates are unfamiliar), vector calculus

emphasized physical/geometrical interpretations, coordinate system fundamentals and some

transformation ideas. In addition, dimensional analysis, Buckingham’s pi theorem, similarity

concepts and non-dimensional differential equations are required to extend concepts further.

Note that only mathematical concepts useful for engineering problem solving and of relevance to

the course on figure 3 are presented. The application areas are also limited due to time

constraints in the course. Motivated students usually learn applications further from this level

through project work and graduate studies.

Figure 4. Organization of Ideal Flow Mathematical Topics

To learn the third level details in figure 4 students must have the necessary background of

differential and integral calculus, differential equations, boundary value problems and concepts

of numerical analysis. The next section presents the connectivity of these to fundamentals of

algebra, geometry and trigonometry beginning from high school.

In this section we review such engineering conceptual strings in relation to mathematics

beginning with high school education. Many mathematical concepts learnt in high school require

earlier nurturing of logical thoughts beginning with arithmetic. Ask students to share their

thoughts about unitary methods, fractions and percentages, and decimals in select groups if

necessary. We shall focus only on definition based abstract thought development beginning with

algebra, geometry and trigonometry to find connectivity later with college calculus. The

recommended approach helps synthesis of concepts and techniques in a meaningful way instead

of presenting mathematics as a subject full of rules and patterns only.

Focus – High School

Listed below are relevant subject areas followed by some discussions of connectivity.

1. Straight Lines – Discuss 3 forms of equations in relation to the origin of coordinates.

2. Connectivity of two points on a straight line to linear interpolation & extrapolation

3. Limiting connections of straight lines to other curves (e.g., secants, tangents, etc.)

Circles, Ellipses, Parabolas & Hyperbolas – specific forms and coordinate origins to

relate focus and directrix in conic sections with eccentricity. – Use suitable wooden or

inexpensive models to demonstrate these.

4. Algebraic ratios, operations with ratios (e.g., simplifying ratios, componendo-dividendo,

etc.) may be introduced to gifted students.

5. Factorization techniques (must be reinforced later in college algebra for root finding of

second and third order curves).

6. Algebraic identities, their connections to factorization and solution of problems with

coordinate geometry.

7. Geometric focus on parallel lines with bisectors and equality of angles, external and

internal angles of a triangle. Focus on right, isosceles, equilateral and similar triangles.

8. Trigonometric focus on definitions in relation to right triangles, sine and cosine laws.

Focus – College

1. Discuss slopes of straight lines & relations that yield connectivity with secants and

tangents to curves. Parametric forms of curves.

2. Discuss matrix Methods of solving algebraic and differential equations.

3. Recall differences of shapes and properties of second order curves.

4. Contrast differential calculus of one independent variable vs. multivariable calculus –

Review ordinary versus substantial derivatives, local and convective accelerations.

5. Emphasize conceptual connectivity of curvilinear coordinates to rectilinear coordinates.

Similarities and differences in unit vectors between rectilinear and curvilinear systems.

6. Review and contrast determination of a streamline by definition with the alternate stream

function approach. Focus on stagnation streamline.

7. Review Taylor series and emphasize on the accuracy and order of terms. Review

engineering approximations and error analysis with examples from later courses.

8. Revive a focus on engineering formulations. Many engineering professors introduce

engineering formulations without any conceptual relevance to mathematics. Instead

formulations must be understood a priori by solvability and uniqueness of solution to

mathematical equations. Disallow all assumptions and let students make them only to

resolve formulation and solvability. Explain earlier used examples from Dynamics and

Fluid Mechanics again to reinforce the ideas.

The above focus in college must be for earlier conceptual recalls and extensions to later

engineering usage. In our view, strengthening dynamics is extremely important for ME students

because all formulations of Fluid Mechanics and ideal flows depend on the extension of rigid

body dynamics concepts [10]. Once basic concepts of moments and force-couple equivalency are

understood, all new learning must be reinforced following these ideas. For example, topics such

as centroids and center of gravity in Statics are excellent to reinforce line, area, and volume

integration concepts. Center of gravity, center of mass, centroids, moments of inertia, parallel

axis theorem, and radius of gyration should not be presented as formulas to be applied, but

conceptually connected as a string of thoughts beginning with their definitions and stages of

simplification. Furthermore, when submerged surfaces are discussed in fluid statics, students

should refrain from using cookbook “I” formulas for location of centroid, force, and moment

calculations. Not only are the integral concepts better, average students would develop advanced

understanding without memorizing use of formulas. Topics such as parallel axis theorem are

required to conceptually connect the Kinetic Diagrams [KD] in Dynamics, and they offer better

understanding of dynamical constraints in motion. Thus, mathematics can be revived, reinforced,

and retained as a powerful synthesizer.

Horizontal and vertical connectivity of each mathematical concept are required with examples

from engineering mechanics concepts. For example, the normal-tangential coordinates are first

introduced in Dynamics but are again used in Fluid Mechanics in defining streamlines. But the

Dynamics course can never have full scope to discuss details of its connectivity. If this topic is

recalled in Fluid Mechanics, an instructor can teach inquisitive students all the complexities in

the governing differential equations due to rotating coordinates. For this purpose, it is important

to discuss in Dynamics how the time derivative of unit vectors is obtained. Many instructors

choose to skip the derivation. This important derivation recalls some geometrical properties of

curves and reviews secants and tangents also. Furthermore, a unique association of magnitude

and directional changes of any vector (e.g., velocity) may be followed later. Once n-t derivation

is completed, ask student teams how they would obtain such unit vector derivatives in r- system

also. Since the angles are measured at the origin of both Cartesian and cylindrical coordinates,

often students mix up the n-t and cylindrical coordinates. The resulting rates of change in angles

must be separately tracked in theses coordinates. This helps (and gets reinforced) during general

motion development in rigid body dynamics later.

The appendix shows 5 home work questions given on the HW set 1 in Fluids II to re-establish

mathematical connectivity as discussed above. For each of these review questions, details of

relevant mathematical recall are discussed individually or, in group mode (depending on the

class size) during office hours. The first question reinforces discussion of a journal bearing

question and the conceptualization of Couette flow in a cylindrical application, where students

use 𝑑𝑢

𝑑𝑦=

𝑟

𝑡. Ask questions like “under what conditions is this linear approximation valid?”,

“what checks would you perform so that the plates may be considered infinite in size compared

to the gap?”, etc. In the second question on HW1 students are urged to review h = a + y sin type

change of variables and choice of an infinitesimal strip to complete the problem rather than using

the “I” formulas from a fluids text. Fluids II would need more integral reviews for flux terms as

depicted on the later questions. The stream function evaluation and velocity profiles are all

relevant recalls for this class. Finally, the last question recalls Reynolds Transport theorem for

mass and momentum theorems on large control volumes. Bulk CV’s set up the reviews on

differential control volumes and derive the differential equations later for interpretation and

understanding of each of the contributing differential terms. Powerful extensions using Taylor

series and order discussions thereof lead to both the finite difference and finite volume methods

in CFD later. The engineering Bernoulli equation which is used in the first fluids course is

brought back in Fluids II to connect both inviscid Bernoulli equation and general convection

equations later after the ideal flow topics. Connectivity is enjoyed by students (who take the Heat

Transfer class concurrently) - knowing the fundamentals from Fluids II, and applying them in

Heat Transfer class simultaneously for engineering designs. Application areas in Fluids II require

understanding of boundary conditions, method of images and inverse design concepts. Some

examples are discussed from each of these areas before velocity and thermal boundary layer

concepts are introduced.

Connectivity – Step 3 (assessment and feedback)

This section presents great success stories quoting benefits of mathematical connectivity. Rarely

we experience any difficulty in implementing the steps mentioned above in a sequence of three

courses – Dynamics, Fluid Mechanics and Fluids II. We present here encouraging performance

data in two specific areas. After studying figure 3, we compare final examination performance on

the use of slopes of curves as we go from Dynamics to Fluid Mechanics. After the introduction

of n-t coordinates, students learn its connectivity to the Cartesian system. The slope of the

velocity vector with the x-axis is the same as the slope of the tangent to the streamline learned

later in Fluid Mechanics. The test performance on two questions from Dynamics are shown with

test performance in Fluid Mechanics on figure 5 below. The first two questions were tested in

2013Spring, 2015Spring and 2016Fall final examinations with grades ranging between 51% and

66% correct respectively, while the last question was tested in Fluid Mechanics in 2017Spring

with 80% of students answering the question correctly.

Also, we tracked the same area of streamlines as we move from Fluid Mechanics to the elective

Fluids II as shown on the figure 6 below. This time the question is related to the topic of

streamline evaluations as shown in the two quiz questions - first as a part of a set of MC

questions, and then as a part of a stand-alone quiz of 10-minute duration on two different times.

On each the class performance was better than 80% correct for the streamline evaluation, plus on

the part (c) related to the ideal flows for the stand-alone quiz.

Figure 6. Streamline concepts carried over from Fluid Mechanics to Ideal Flows

Note that in the above two questions, the MC question asks for the details to get a full credit. The

work load is not much beginning with the equation (available on the formula sheet) 𝑑𝑥

𝑢=

𝑑𝑦

𝑣.

The important task is to check if students carry out the integration and the constant evaluation

correctly by fitting the point (2, 1) on the streamline. If they do not show the work, they get only

one point, usually allocated for MC questions on a class quiz. They may also incorrectly perform

the integration, simplify the ratio incorrectly, or, evaluate the constant incorrectly, which would

be revealed from choices. If the class performance was not up to expectations, students would

still get the opportunity to improve their score on the question by resubmitting the quiz. Details

of the assessment leading to connectivity of ideal flow topics are discussed in references [3 - 5].

If the class size is small enough, alternate and accurate assessment may be performed using a

stand-alone quiz as the Quiz No. 3 shows. As always, a prompt return of the original quiz is

necessary so that students can relearn the missed topic over the approaching weekend.

Similar to the above examples, the focus area of vectors was traced on examples from Dynamics

to Fluid Mechanics. In the absence of tensors, vectors are the primary focus for our ME students

and the coverage is very broad beginning with coordinate systems, unit vectors, inner and outer

products, vector calculus, etc. We have tested many detailed areas of importance individually,

with positive results demonstrated due to reinforcement and connectivity. Here one such sample

is shown on figure 7 below. For the first two questions taken from Dynamics, the focus is on

component addition/subtraction to obtain relative velocity, but the second question uses the

geometrical construction of a negative vector and parallelogram/triangle of vectors. The third

example on the figure is from fluid mechanics involving dot products and its interpretation in

finding fluxes of fluid momentum. Once again, the performance was better than 65% correct on

each examination demonstrating that such instructional reinforcement is very effective (because

ESCC considers 60% correct class performance on each MC question as acceptable).

The mastery in ideal flow topics may further be traced using breadth and depth of our student

interests. Two students took the initiative to prepare a set of nicely typed 550CD notes (Note:

Fluids II used to be taught as EMEM550 – Transport Phenomena when RIT was on a quarter

system) to be shared later with future batches. Two projects with papers were developed, both of

which published a user-friendly MATLAB interactive environment for learning of ideal

flows/aerodynamics [11]. Another student prepared a journal paper for the use of an advanced

learning platform to prepare high accuracy CFD codes to study the modified equation approach

for solving hyperbolic partial differential equations [12]. But these are highly motivated students

that truly understood the approaches. For struggling students, even the rigid body general motion

questions in dynamics are not that straightforward. Therefore, various types of reinforcement

are necessary. At this time more faculty participation is necessary to fix such learning

bottlenecks.

We close the present discussion with one example which illustrates how much effort is currently

spent in exploring possible connectivity of mathematical topics. In a mechanical engineering

program, students have an added advantage of observing applied examples and verifying

mathematical models by experiments. Beginning with Statics, mathematical traces are recalled

all the way to upper level courses such as ideal flows. Dynamics receives a pivotal importance

for this purpose. This final example also points out a new trend [1] that is developing in student

performance (which defies our concerted planning efforts).

One of the difficult conceptual areas that students should master at the end of Dynamics is the

area of rigid body rotation. In deformable media, rotations are more complex and shear and

rotational contributions are coupled by the same derivatives in planar flows. While vorticity

𝑧 = (𝑣

𝑥−

𝑢

𝑦) 𝑎𝑛𝑑, 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑦𝑥 = (

𝑣

𝑥+

𝑢

𝑦) are taught during the first weeks in flow

kinematics, more focus on the vortex motion may be desirable leading to aerodynamic and

meteorological applications. Many questions of varying complexity may be constructed. Thus,

Dynamics has a major focus in this area beginning with the instantaneous center of zero velocity

[1], [6], [7].

We present below three morphed attempts from a focus question in Dynamics (Figure 8) and

summarize its performance on a bar chart (Figure 9). About six years ago, a new topic of KD had

newly been included in Dynamics syllabus by the ESCC faculty in addition to the usual Free

Body Diagram (FBD). This particular example has two separate objects with mass (a disk and a

block) but the mass of the disk is not provided explicitly in the question. Also, the disk is pinned

at the center not allowing it to translate. When the block is released from rest, net torque using

KD must include the moment due to the block’s inertia force about the center, and add to it the

moment due the couple I0α. In spite of a constant focus by 3 different ESCC teams from 2104 to

2018, the performance seems to have deteriorated after 2015.

Figure 8. Focus question from a recent Dynamics final exam (see performance in figure 9)

Figure 9. Percent correct and largest incorrect responses on the question in figure 8

Note that the question on figure 8 was morphed three times. In the first two years (2014 & 2015)

the question simply asked to compute (Mk)0 as obtained from the KD in terms of the angular

acceleration of the disk α. The choices given were:

(A) 0.005α, (B) 0.015 α, (C) 0.1α, (D) 0.01 α, and (E) 0 N-m.

For this version of the test, the correct choice was (B) and the largest incorrect choice given by

students was (A). Therefore, on the figure 9 legend, replace the choices (D) and (E) by (B) and

(A), respectively to make correct comparisons. The topic of KD was new on the syllabus, noted

from [13]. So, the first response in 2014-Spring was understandable. After specially focusing on

this area, the question was re-tested in fall semester final examination of 2015, when the

responses improved to 44% correct, but still the incorrect answer (A) was chosen by 42%

students. Incidentally, the choice (A) would be obtained if the mass of the block was neglected

altogether. In three following final examinations after that, the question was posed in the current

version (Fig. 8) but without the hint that is displayed on the question now. The correct and

largest incorrect choices now are (D) and (E) respectively. After noticing not much

improvement in correct responses, the question was posed in the present form (with the added

hint) in 2018-Fall semester. However, the performance never seems to improve much even with

the hint (since our target achievement level of 60% of the class being correct was never

reached). The incorrect response (E) comes from neglecting the disk altogether as if the block

fell independently under gravity. There may be several reasons for this lack of performance.

Interviewing students after the test in 2018 revealed that they expected an easier question. Many

students did not even read the hint. We plan to attempt using other means of reinforcement for

this question in future. We have presented other examples and discussion of connectivity with

engineering concepts in another publication in this conference [14].

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

2014-Spring 2015-Fall 2016-Spring 2017-Fall 2017-Spring 2018-Fall

% Correct Response and Largest Incorrect Response

ChoiceD ChoiceE

To summarize, the current trend of student performance reflects neglect of important

mathematical concepts, and answering without proper technical considerations. If this trend

persists for long irrespective of our instructional efforts, subjects requiring more in-depth

deliberations would be difficult to deliver. Unable to recall relevant concepts required to solve a

question during an examination, students usually guess or reply using a layman’s approach [2]. It

is therefore the responsibility of engineering educators to reinitiate interests and emphasize

logical analysis without which applied mathematics may not be appreciated. We now know that

poor performance in the question on figure 1 was due to erroneous integration methods used.

Hopefully the methods and connectivity which were demonstrated here with ideal flow topics

would be the starting point of a new instructional design for continuous connectivity of topics.

Conclusion

This paper presented mathematics as a common bond in developing ME thoughts. The approach

seeks connectivity of mathematical topics/concepts to teach students engineering formulations

related to ideal flows and CFD. This effort was necessary after faculty observed performance in

ESCC examinations on mathematics related topics fell short of expectations. Positive results of

applying the present approach were reported here for ideal flow applications. New solutions will

be explored in future to address the challenges mentioned here.

Acknowledgments

This study would not have been possible without active participation of all members of various

ESCC instructional teams for the past twelve years. All participating faculty members and

trained teaching assistants helped in designing, discussing, and evaluating examination archives

from which examples are presented here. The author, who was also the ESCC coordinator in the

past, gratefully acknowledges all their contributions. The author would also like to thank Drs. E.

C. Hensel and R. Robinson for helpful discussions and support.

Appendix

The following home assignment set is designed to review several mathematical procedures and

concepts for connectivity of Fluids II with previous courses.

References

[1] A. Ghosh, “Formative Assessment using Multiple Choice Questions in Statics and

Dynamics”, Paper No. IMECE-66304, ASME International Mechanical Engineering Congress

and Exposition, November 11 – 17, Phoenix, Arizona, 2016.

[2] A. Ghosh, “Foundations of Statics – An Assessment Study and Feedback Implementation”,

Paper No. IMECE-66302, ASME International Mechanical Engineering Congress and

Exposition, November 11 – 17, Phoenix, Arizona, 2016.

[3] A. Ghosh, “Motivation Building Strategies of Mathematics Instruction for Undergraduate

Students in Mechanical Engineering”, Paper ID #23730, ASEE Annual Conference and

Exposition, Salt Lake City, Utah, 2018.

[4] A. Ghosh, "Teaching Formulation Skills in an Upper Level Fluid Mechanics Course", Paper

No. IMECE-63989, ASME International Mechanical Engineering Congress and Exposition,

November 11 – 17, Denver, Colorado, 2011.

[5] A. Ghosh, “Development of Analytical Skills through Cooperative Learning”, Paper No.

IMECE-12947, ASME International Mechanical Engineering Congress and Exposition,

November 13 – 19, Lake Buena Vista, Florida, 2009.

[6] A. Ghosh and E.C. Hensel, “An interpretive assessment of engineering science core courses”,

Paper No. IMECE-12939, ASME International Mechanical Engineering Congress and

Exposition, November 13 – 19, Lake Buena Vista, Florida, 2009.

[7] A. Ghosh, "Use of Multiple-Choice Questions as an Assessment Tool in Dynamics", Paper

No. IMECE-63987, ASME International Mechanical Engineering Congress and Exposition,

November 11 – 17, Denver, Colorado, 2011.

[8] R. C. Hibbeler, Fluid Mechanics, Pearson Education, 2018.

[9] A. H. Shapiro, “Vorticity – Parts I & II”, National Committee for Fluid Mechanics Films,

MIT Press, 2008.

[10] A. Ghosh, “Analysis of a Feedback Assessment Loop in Engineering Sciences Core

Curriculum”, Paper IMECE-66486, International Mechanical Engineering Congress &

Exposition, San Diego, California, USA, Nov. 15-21, 2013.

[11] A. Ghosh, and C. Pantaleon, “Teaching Computational Fluid Dynamics Using MATLAB”

Paper IMECE-66458, International Mechanical Engineering Congress & Exposition, San Diego,

California, USA, Nov. 15-21, 2013.

[12] C. Pantaleon, and A. Ghosh, “Taylor series expansion using matrices: an implementation in

MATLAB”, Computers & Fluids, Volume 112, 2 May 2015, Pages 79–82, 2015.

[13] R. C. Hibbeler, Engineering Mechanics – Dynamics, 13th Edition, Pearson, 2013.

[14] A. Ghosh, ASEE Annual Conference and Exposition, Tampa, Florida, 2019 (in preparation)